In order to carry out parametric tests, there are several assumptions that need to be satisfied and one of the main ones is the normality assumption. If the data are not normally distributed, it is better to simply opt for the non-parametric equivalent for t test and ANOVA such as the Kruskal Wallis Test.

Here's an article where it has been used instead of ANOVA/t-test.

Article Wellbeing of Palliative Care Workers During Covid-19 Pandemi...

Thanks s. Rama Gokula Krishnan

@s. Rama Gokula Krishnan

"If the data are not normally distributed..." this is not an assumption of anova. Sources that say so should be disregarded.

I recommend starting with a credible source. For the assumptions of a general linear model, including anova, for example, see the following presentation, slide 8, and then slides 19 - 22. Taken from Montgomery, Design and Analysis of Experiments.

https://staff.emu.edu.tr/adhammackieh/Documents/courses/ieng581/lecture-notes/ch03.pdf

Sharon Xuereb ,

I won't this a comprehensive post, but I'll make a few comments.

• Testing for heteroscedasticity as pre-condition for another analysis isn't helpful. With a relatively large sample size, Levene's test may return a significant result, even when the heteroscedasticity is mild. It's much better to plot residuals to assess the heteroscedasticity to see if it's "reasonable enough" for the analysis you are considering.
• But if you're talking about t-tests, you can just use the t-test for unequal variances. So, heteroscedasticity may not be a concern anyway.
• What parametric statistics are you using ? What's the design of the study ? It sounds like you are using a bunch of t-test, when a single model may be a better approach.
• I'd be cautious about removing outliers. It's rarely justified, unless the data is actually wrong.

Sal Mangiafico thanks so much for your thoughts and the resource. I'm using ANOVAs (it could be t-tests, but with ANOVAs I believe I reduce the risk of false positives).

I'm now wondering ... if I can use the unequal variances assumed option for ANOVAs, then when would a non-parametric test be needed?

For your info, I asked participants to answer questions about how they perceive refugees from Syria and from the Ukraine, and I want to compare, to examine if Ukrainians are perceived more favourably due to my participants being European.

Thanks!

Hi, Sharon Xuereb .

Yes, a one-way anova can be adjusted for heteroscedasticity. There's Welch's anova, which should be equivalent to Welch's t-test for two groups. (Welch's t-test is the t-test commonly used when with unequal variances.)

There are other approaches like White adjustment (sandwich estimators). There's a whole discussion here on different approaches to the problem: https://stats.stackexchange.com/questions/91872/alternatives-to-one-way-anova-for-heteroskedastic-data .

On nonparametric tests, I think the best approach is to realize that nonparametric tests test different hypotheses than traditional parametric tests (like anova). Sometimes those hypotheses are the ones we are interested in. And sometimes they're just a simple approach to answer the question.

I don't really agree with the idea that we should use e.g. anova unless we can't for some reason.

Usually if we select an appropriate for the kind of data we have, and can make reasonable assumptions about the necessary assumptions, the model should be reasonably valid. (There are models for count data, binomial data, left-limited skewed data, and so on).

But if we take the approach that we should use e.g. anova unless the assumptions aren't met... Really it comes down to if the residuals are reasonably normal, and if there is some reasonable way to account for heteroscedasticity. If not, it may be easiest to use a nonparametric approach.

Thanks so much!

Sal Mangiafico The measurement variable has to be normally distributed or else it might lead to false positives, even in the case of large samples. This has already been posted on ResearchGate and answered by experts.

https://www.researchgate.net/post/To-what-extent-it-is-necessary-to-verify-the-normality-of-the-data-in-ANOVAs-analysis

s. Rama Gokula Krishnan , no, the idea that the measurement variable is normally distributed is absolutely not an assumption of e.g. anova.

Honestly, I would ask you look at an analysis of experiments textbook and see what the assumptions are. The link I shared above is a fair description of the normality assumption.

Unfortunately there is a lot of misinformation about this. Certainly over the decades I've been involved in analysis of experiments. Even among those who teach data analysis or write relevant websites.

Mostly, it is just sloppy language, saying "the data should be normally distributed" or "normality assumption" without explaining what the assumption is.

To be clear:

On the one hand, the assumption is not about the data.

But also, the assumption is not about the distribution of the dependent variable, but about the distribution of the errors --- usually estimated by the residuals of the analysis.

Practically speaking, think about a simple example of two groups with distinctly different means of the dependent variable. If this is the case, of course the dependent variable wouldn't be normally distributed. Under the assumption of normally distributed data, t-tests and anova would be worthless in any meaningful case. Consider the image: https://rcompanion.org/handbook/images/image095.png

It seems to be a language-problem (linguistic? problem) here.

The crucial point is Y|x ~ N(µ(x), σ²) where µ(x) is some function of the predictor values (x). If one considers one specific set of conditions (one specific x), one can omit it to get Y ~ N(µ, σ²) where the µ is just a constant.

The formula can be re-written as y = µ(x) + e, where y is an observed value (a realization) of Y and e is a residual, which can be seen as an observation/realization of a random variable E ~ N(0, σ²).

It seems cumbersome to me to fight whether Y, Y|x or E is normal distributed. In an experiment where x plays the major role (as it represents the experimental conditions that are supposed to have an impact on µ) it should be obvoius that the distributional assumptions consider that µ may depend on x.

If one says that Y is normal distributed does not neccesarily imply that this normal distribution must have a specific mean and a specific variance. Of course may the mean depend on experimental conditions (this is why we do the experiment). But this does not invalidate the fact that Y is normal distributed. We only assume that Y|x1 and Y|x2 have the same mean when x1=x2. Just when takling about E, the impact of x on the mean is "removed" (via subtraction), so E is normal distributed with mean 0, independent of x.

Jochen Wilhelm , I agree. However, there's a practical point, which is that people on forums like this often look at --- or more often test --- their dependent variable for normality. And finding it not normal, conclude that they cannot use traditional parametric approaches like anova. I've seen it plenty.

This is obviously a widespread misunderstanding. I do think it's often perpetuated by imprecise language, saying, for example, "The data need to be normally distributed."

If you'll indulge me, I've attached a better example to make the point explicit for readers --- taken from: https://elmscontent.dot.gov:8080/elms/faa/ato/to/hfwebtraining/HFModel/Variance/anthropometrics2.htm.

(This is a histogram of the height of men in the U.S. Air Force and women in the U.S. Women in the Air Force, 1967 and 1968 respectively.)

Presumably using a t-test would be appropriate for these data. The histograms of the individual groups look approximately normal, and we know that, as a population, height in people is approximately normal.

But if we look at the dependent variable in the combined groups, their distribution wouldn't be normal. I think it would be distinctly bi-modal.

We can imagine that if we had three groups like this, anova would be appropriate. But if we look at the distribution of the dependent variable, it would clearly be non-normal.

Yep, the mistake (imprecise language) already starts with "the data need to be normally distributed". The data is just a collection of fixed values. It might have a frequency distribution, and there is no need that this frequency distribution is a good approximation to the normal probability distribution. Rather, each and every single value should be resonably assumed to be a realization of a random variable that has a normal distribution. If all these random variables also share their mean and their variance, then the frequency distribution of a large sample should (is expected to) approximate the normal distribution specified by this mean and this variance (what may be visualized and "tested", as you wrote).

This bad/sloppy/wrong/impercise use of language leads to may misunderstandings and hinders many to think further and apply better analyses to their data. It is interesting that many people don't seem to have a problem to understand that a variable like "number of females in clutch of size n" is binomial-(n,p) distributed, certainly being aware that the specific distribution depends on the value of p and that p is supposed to depend on experimental conditions. So Y|x ~ Bin(n, µ(x)), again. In contrast to the normal distribution, Y - µ(x) is not binomial distributed (not sure what the distribution is, but is does not really matter anyway, imo).

It took me so many years to wrap my head around this because the literature is full of such bad examples. This is why I take the effort here at the risk of looking nit-picky.

Thank you so much for this discussion, Sal Mangiafico and Jochen Wilhelm . This has really opened my eyes re this important subject!

There are a few things to consider when deciding whether to use non-parametric statistics:

• The type of data you have.Parametric statistics require that your data be continuous and normally distributed. Non-parametric statistics can be used with any type of data, including categorical, ordinal, and non-normally distributed data.
• The size of your sample. Parametric statistics are generally more powerful than non-parametric statistics, but they require a larger sample size to be reliable. Non-parametric statistics can be used with smaller sample sizes.
• The assumptions of the parametric test. Parametric tests make certain assumptions about the data, such as normality and homogeneity of variance. If these assumptions are not met, then the results of the parametric test may be unreliable. Non-parametric statistics do not make these assumptions, so they can be used even if the assumptions of the parametric test are not met.

Here is a table that summarizes the key advantages and disadvantages of parametric and non-parametric statistics:

FeatureParametricNon-parametricPowerMore powerfulLess powerfulSample size requirementLarger sample size requiredSmaller sample size requiredAssumptionsRequires normality and homogeneity of varianceDoes not require normality or homogeneity of variance

In general, it is best to use parametric statistics if you have continuous, normally distributed data and a large enough sample size. However, if you have categorical, ordinal, or non-normally distributed data, or if your sample size is small, then non-parametric statistics are a better choice.

Here are some examples of when you might want to use non-parametric statistics:

• You are comparing two groups of people based on their income, which is categorical data.
• You are comparing two groups of students based on their grades on a test, which is ordinal data.
• You are comparing two groups of people based on their height, which is continuous data but not normally distributed.
• You have a small sample size, regardless of the type of data.

If you are unsure whether to use parametric or non-parametric statistics, it is always best to consult with a statistician.

Thanks Ahlam Hanash Gatea

Should be "Thanks to ChatGPT"... Ahalm did just copy&paste.

Nina Nikolic